This problem is not "well posed". If we asume that the set of n numbers can include negative numbers, then there is no largest possible number. I am going to assume that the ten "numbers" are, in fact, positive integers.

Call the largest number "n". Suppose 15 occurs eight times so the mode is 15. We would need to have 1 numbers less than 15 and 1 number, n, larger than 15 so that median is also 15. The smallest positive integer would be 1. In order that the mean be 15 we would have to have 1+ 8(15)+ n= 10(15) so that n= 2(15)- 1= 29.

If we assume that the 10 numbers can be any non-negative numbers, then the smallest number in the set would be 0 and we would have to have 0+ 8(15)+ n= 10(16) so that n= 2(15)= 30.